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The Euler equations as a RNG fixed point

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In this paper at the at the beginning of the last paragraph on p.2 it is said, that the Euler equations, which are an infinite Reynolds number limit of the Navier-Stokes equations, arise as an RNG fixed point. This fixed point is said to be non-unique for a system witn N mixing species, because it can be choosen from an N+1 dimensional parameter space spanned by N-1 dimensionless mass diffusivities, the dimensionless thermal diffusivity (or Prantl Number) and the dimensionless ratio of the anisotropic and isotropic viscosity.

About this issue I have several closely related questions:

What kind of fixed point is this when considering a classification into Gaussian/interacting, trivial/nontrivial, etc fixed points

What is the expected behaviour of the RG flow around this fixed point?

Is the non-uniqueness of this fixed point the same kind of non-uniquenes as the one described on p67 if this paper, which explains that the presence of redundant marginal operators can lead to a whole line of physically equivalent fixed points? If this way of analyzing a fixed point can be applied in my example, would the Euler equations fixed point then correspond to some kind of an N+1 dimensional surface of fixed points where the mass diffusivities, the prantl number, and the ratio of the anisotropic and isotropic viscosity play the role of such redundant marginal operators?

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1 Answer

Ok, here is how I see these issues now after taking a bit more time to think about them:

In the first paper it is said that the Euler equations emerge as the infinite Reynolds number limit of the Navier Stokes equation which means that according to the definition of the Reynolds number

$$ RE = \frac{V L}{\nu} $$

the molecular diffusion can be neglected in this case. As is known that to fully developped turbulence all scales are expected to conribute (there is no characteristic scale present) and molecular diffusion can be neglected at larg scales, the fixed point corresponding to the Euler equations from a RG point of view is a critical IR fixed point. However as mentioned here when looking at LES dynamic subgrid scale parameterizations from a RG point of view, talking of a (scale invariant) fixed point is not exactly justified because the rescaling is missing in the renormalization step, and the IR limit the system approaches when repeating this modified renormalization step is more exactly called a limit point.

Looking at the non-uniqueness of this fixed (or limit) point mentioned in the first paper from the point of view explained around p. 67 of the second paper, this non-uniqueness does not mean that the N+1 parameters span some kind of a higher dimensional generalization of a line of fixed points, as the corresponding operators are relevant instead of redundant and marginal. What is meant instead, is when doing a linear analysis of the RG flow around the fixed (limit) point such that the nearby action is given by

where $S_{*}(\phi)$ is the action at the fixed (relevant) point and $\alpha_i$ are integration constants, there are N+1 relevant operators $O_i(\phi)$ for a system with N mixing species with Eigenvalues $\lambda_i > 0$ which are determined from the Eigenvalue equation

$$ M O_i(\phi) = \lambda_i O_i(\phi) $$

The non-uniqueness alluded to in the first paper corresponds to the fact that the integration constants $\alpha_i$ are not determined by the renormalization procedure itself but as explained in the second paper have to be determined by the bare action or perfect action which lies on a renormalized trajectory.

In the context of Large Eddy Simulations (LES) that make use of dynamic subgrid scale parameterizations for turbulent diffusion for example, it is possible to dispense with the non-uniqueness by calculating the corresponding integration constant directly from the resolved scale by making use of the Germano identity Eq. (4.2) in the second paper and application of the Smagorinsky scheme to calculate a dynamic mixing length.

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